The Annals of Probability

Poly-logarithmic localization for random walks among random obstacles

Jian Ding and Changji Xu

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Place an obstacle with probability $1-\mathsf{p}$ independently at each vertex of $\mathbb{Z}^{d}$, and run a simple random walk until hitting one of the obstacles. For $d\geq2$ and $\mathsf{p}$ strictly above the critical threshold for site percolation, we condition on the environment where the origin is contained in an infinite connected component free of obstacles, and we show that the following path localization holds for environments with probability tending to 1 as $n\to\infty$: conditioned on survival up to time $n$ we have that ever since $o(n)$ steps the simple random walk is localized in a region of volume poly-logarithmic in $n$ with probability tending to 1. The previous best result of this type went back to Sznitman (1996) on Brownian motion among Poisson obstacles, where a localization (only for the end point) in a region of volume $t^{o(1)}$ was derived conditioned on the survival of Brownian motion up to time $t$.

Article information

Ann. Probab., Volume 47, Number 4 (2019), 2011-2048.

Received: September 2017
Revised: July 2018
First available in Project Euclid: 4 July 2019

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Mathematical Reviews number (MathSciNet)

Primary: 60K37: Processes in random environments 60H25: Random operators and equations [See also 47B80] 60G70: Extreme value theory; extremal processes

Random walk among random obstacles localization


Ding, Jian; Xu, Changji. Poly-logarithmic localization for random walks among random obstacles. Ann. Probab. 47 (2019), no. 4, 2011--2048. doi:10.1214/18-AOP1300.

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  • [1] Antal, P. (1995). Enlargement of obstacles for the simple random walk. Ann. Probab. 23 1061–1101.
  • [2] Antal, P. and Pisztora, A. (1996). On the chemical distance for supercritical Bernoulli percolation. Ann. Probab. 24 1036–1048.
  • [3] Astrauskas, A. (2007). Poisson-type limit theorems for eigenvalues of finite-volume Anderson Hamiltonians. Acta Appl. Math. 96 3–15.
  • [4] Astrauskas, A. (2008). Extremal theory for spectrum of random discrete Schrödinger operator. I. Asymptotic expansion formulas. J. Stat. Phys. 131 867–916.
  • [5] Athreya, S., Drewitz, A. and Sun, R. (2017). Random walk among mobile/immobile traps: A short review. Preprint. Available at arXiv:1703.06617.
  • [6] Biskup, M. and König, W. (2016). Eigenvalue order statistics for random Schrödinger operators with doubly-exponential tails. Comm. Math. Phys. 341 179–218.
  • [7] Biskup, M., König, W. and dos Santos, R. S. (2018). Mass concentration and aging in the parabolic Anderson model with doubly-exponential tails. Probab. Theory Related Fields 171 251–331.
  • [8] Bolthausen, E. (1994). Localization of a two-dimensional random walk with an attractive path interaction. Ann. Probab. 22 875–918.
  • [9] Chayes, J. T., Chayes, L. and Newman, C. M. (1987). Bernoulli percolation above threshold: An invasion percolation analysis. Ann. Probab. 15 1272–1287.
  • [10] Donsker, M. D. and Varadhan, S. R. S. (1975). Asymptotics for the Wiener sausage. Comm. Pure Appl. Math. 28 525–565.
  • [11] Donsker, M. D. and Varadhan, S. R. S. (1979). On the number of distinct sites visited by a random walk. Comm. Pure Appl. Math. 32 721–747.
  • [12] Fiodorov, A. and Muirhead, S. (2014). Complete localisation and exponential shape of the parabolic Anderson model with Weibull potential field. Electron. J. Probab. 19 no. 58, 27.
  • [13] Fukushima, R. (2009). From the Lifshitz tail to the quenched survival asymptotics in the trapping problem. Electron. Commun. Probab. 14 435–446.
  • [14] Gärtner, J., König, W. and Molchanov, S. (2007). Geometric characterization of intermittency in the parabolic Anderson model. Ann. Probab. 35 439–499.
  • [15] Gärtner, J. and Molchanov, S. A. (1990). Parabolic problems for the Anderson model. I. Intermittency and related topics. Comm. Math. Phys. 132 613–655.
  • [16] Grenkova, L. N., Molčanov, S. A. and Sudarev, J. N. (1983). On the basic states of one-dimensional disordered structures. Comm. Math. Phys. 90 101–123.
  • [17] Grimmett, G. R. and Marstrand, J. M. (1990). The supercritical phase of percolation is well behaved. Proc. Roy. Soc. London Ser. A 430 439–457.
  • [18] Hall, R. R. (1992). A quantitative isoperimetric inequality in $n$-dimensional space. J. Reine Angew. Math. 428 161–176.
  • [19] Kesten, H. and Zhang, Y. (1990). The probability of a large finite cluster in supercritical Bernoulli percolation. Ann. Probab. 18 537–555.
  • [20] König, W. (2016). The Parabolic Anderson Model: Random Walk in Random Potential. Birkhäuser/Springer, Cham.
  • [21] König, W., Lacoin, H., Mörters, P. and Sidorova, N. (2009). A two cities theorem for the parabolic Anderson model. Ann. Probab. 37 347–392.
  • [22] Lacoin, H. and Mörters, P. (2012). A scaling limit theorem for the parabolic Anderson model with exponential potential. In Probability in Complex Physical Systems. Springer Proc. Math. 11 247–272. Springer, Heidelberg.
  • [23] Lawler, G. F. and Limic, V. (2010). Random Walk: A Modern Introduction. Cambridge Studies in Advanced Mathematics 123. Cambridge Univ. Press, Cambridge.
  • [24] Lee, S. (1997). The power laws of $M$ and $N$ in greedy lattice animals. Stochastic Process. Appl. 69 275–287.
  • [25] Martin, J. B. (2002). Linear growth for greedy lattice animals. Stochastic Process. Appl. 98 43–66.
  • [26] Povel, T. (1999). Confinement of Brownian motion among Poissonian obstacles in ${\mathbf{R}}^{d},d\ge3$. Probab. Theory Related Fields 114 177–205.
  • [27] Sidorova, N. and Twarowski, A. (2014). Localisation and ageing in the parabolic Anderson model with Weibull potential. Ann. Probab. 42 1666–1698.
  • [28] Sznitman, A.-S. (1990). Lifschitz tail and Wiener sausage. I, II. J. Funct. Anal. 94 223–246, 247–272.
  • [29] Sznitman, A.-S. (1991). On the confinement property of two-dimensional Brownian motion among Poissonian obstacles. Comm. Pure Appl. Math. 44 1137–1170.
  • [30] Sznitman, A.-S. (1993). Brownian asymptotics in a Poissonian environment. Probab. Theory Related Fields 95 155–174.
  • [31] Sznitman, A.-S. (1993). Brownian survival among Gibbsian traps. Ann. Probab. 21 490–508.
  • [32] Sznitman, A.-S. (1996). Brownian confinement and pinning in a Poissonian potential. I, II. Probab. Theory Related Fields 105 1–29, 31–56.
  • [33] Sznitman, A.-S. (1997). Fluctuations of principal eigenvalues and random scales. Comm. Math. Phys. 189 337–363.
  • [34] Sznitman, A.-S. (1998). Brownian Motion, Obstacles and Random Media. Springer, Berlin.
  • [35] van der Hofstad, R., Mörters, P. and Sidorova, N. (2008). Weak and almost sure limits for the parabolic Anderson model with heavy tailed potentials. Ann. Appl. Probab. 18 2450–2494.
  • [36] Whittington, S. G. and Soteros, C. E. (1990). Lattice animals: Rigorous results and wild guesses. In Disorder in Physical Systems. 323–335. Oxford Univ. Press, New York.